Adaptive Stabilization For Several Classes Of Uncertain Distributed Parameter Systems

This dissertation focuses on the investigation of the stabilization for sev-eral classes of uncertain systems including of uncertain heat equations, ODE sys-tems with uncertain diffusion-dominated actuator dynamics and uncertain coupled PDE-ODE systems. Moreover, the dissertation also investigates the state feed-back and output feedback stabilization for a class of coupled PDE-ODE systems with spatially varying coefficient. For details, the main content of the dissertation consists of the following four parts:(Ⅰ) Adaptive stabilization for a class of heat equationsThis part is Chapter2of the dissertation, and studies the adaptive stabiliza-tion for a class of heat equations with uncertain control coefficient and boundary disturbance. By using Lyapunov direct method, the adaptive state feedback con-troller is explicitly constructed, which guarantees that all the closed-loop system states are bounded while the original system states being L2stable. Particularly, the original system states converge to zero while the boundary disturbance van-ishing. It is worthwhile to point out that, the designed controller only requires the measurements at one end of the heat equation, and hence reduces the burden of measurement in the existing literature. Moreover, by skilfully choosing initial condition of the parameter updating law, the restriction on the initial conditions of the system is moderately relaxed, which is usually described by the so-called compatible condition in the existing literature.(Ⅱ) Adaptive stabilization for ODE systems with uncertain diffusion-dominated actuator dynamicsThis part is Chapter3and4of the dissertation. Chapter3considers the adaptive state feedback stabilization for ODE systems with uncertain diffusion-dominated actuator dynamics. Comparing to the related literature, the problem under investigation is more difficult to solve due to the presence of the serious un-certainties. By introducing an infinite-dimensional backstepping transformation, a pivotal target system is firstly obtained, which makes the control design and per-formance analysis of the closed-loop system more convenient. Then, based on the certain equivalence principle and other developed adaptive techniques, an adaptive state feedback controller is designed, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero. Chapter4studies the adaptive output feedback stabilization for ODE systems with uncer-tain diffusion-dominated actuator dynamics. Different from the last chapter where all the states of the actuator dynamics are measurable, only one of the bound-ary values of the actuator is available for feedback, and hence makes the control design much difficult. By constructing state observers, the estimations of the un-observable states are first obtained, then by the infinite-dimensional backstepping method and certain equivalence principle, the adaptive controller is constructed, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero.(Ⅲ) Stabilization for coupled PDE-ODE systems with spatially varying coefficientThis part is Chapter5of the dissertation, and investigates the stabilization of a class coupled PDE-ODE systems with spatially varying coefficient via both state feedback and output feedback. The system under consideration is more gen-eral than that of the related literature due to the presence of the spatially varying coefficient which makes problem more difficult to solve. By infinite-dimensional backstepping method, both state feedback and output feedback controllers are designed, which guarantee that the closed-loop system is exponentially stable in certain sense of norm. It is worth pointing out that, in the cases of the output feed-back, the restriction on the ODE subsystem in the existing results is completely removed by choosing appropriate state observer gains.(Ⅳ) Adaptive stabilization for uncertain coupled PDE-ODE sys-temsThis part is Chapter6and7of the dissertation, and focuses on the adaptive stabilization for two classes of uncertain coupled PDE-ODE systems. Chapter6studies the adaptive stabilization for uncertain coupled PDE-ODE system with only one unknown parameter. The presence of the uncertainties makes the sys-tem under investigation essentially different from those in the existing literature, and results in the incapability of the existing methods. Motivated by the related works, adaptive controller is designed by using the infinite-dimensional backstep-ping method and certain equivalence method, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero. Differently, Chapter7considers the adaptive stabilization for a class of uncer-tain coupled PDE-ODE systems with multiple unknown parameters, particularly there are unknown parameters exist in the ODE sub-system. Thus, the system under consideration is much general and much difficult to control. By the existing methods, particularly the infinite-dimensional backstepping method, the adap-tive controller is designed, which guarantees all the closed-loop system states are bounded while all the original systems states converging to zero.For the theoretical results obtained in the above four parts, the corresponding simulation examples are presented to illustrate the effectiveness of the proposed methods.